Faulty calculations of the “value” that D.C. teachers added to student achievement in the last school year resulted in erroneous performance evaluations for 44 teachers, including one who was fired because of a low rating, school officials disclosed Monday.

School officials described the errors as the most significant since the system launched a controversial initiative in 2009 to evaluate teachers in part on student test scores.

Half of the evaluations for the 44 teachers were too high and half too low, said Jason Kamras, chief of human capital for D.C. Public Schools.

Those affected are about 1 percent of about 4,000 teachers in the school system. But they comprise nearly 10 percent of the teachers whose work is judged in part on annual city test results for their classrooms.

Kamras said the school system will leave unchanged the ratings that were too high and will raise those that were too low. He said the school system is seeking to reinstate the fired teacher and will compensate the teacher — whose identity was not revealed — for lost salary.

“We will make the teacher completely whole,” he said.

In addition, Kamras said, three teachers whose ratings are being revised upward will shortly receive bonuses of $15,000 each.

The evaluation errors underscore the high stakes of a teacher evaluation system that relies in part on standardized test scores to quantify the value a given teacher adds to the classroom. The evaluation system, known as IMPACT, has drawn widespread attention since it began under former D.C. schools chancellor Michelle A. Rhee. It remains a centerpiece of efforts to raise the performance of long-struggling schools in the nation’s capital — and a flashpoint in the national school-reform debate.

Backers of IMPACT say it is essential to hold ineffective teachers accountable for poor results and reward those who are highly effective. Critics say efforts to distill teaching outcomes to a set of numbers are misguided and unfair.

Elizabeth A. Davis, president of the Washington Teachers’ Union, said the disclosure of mistaken teacher ratings for the 2012-13 school year was disturbing.

“IMPACT needs to be reevaluated,” Davis said. “The idea of attaching test scores to a teacher’s evaluation — that idea needs to be junked.”

Davis sent a letter to D.C. Schools Chancellor Kaya Henderson demanding more information about the errors and the evaluation system.

Kamras said school officials moved to rectify the errors as soon they learned of them from Mathematica Policy Group, the research firm the city hired to crunch numbers used in the evaluations.

“We take these kind of things extremely seriously,” Kamras said. “Any mistake is unacceptable to us.”

The value-added calculations are complex. The first step is to estimate how a teacher’s students are likely to perform on the citywide D.C. Comprehensive Assessment System, based on past test results and other information. Then the predicted classroom average is compared to the actual classroom average. The difference is what school officials call the value that a teacher adds.

The value-added formula applies to English language arts teachers in grades four through 10 and to math teachers in grades four through eight — about 470 instructors in all. Kamras said the faulty calculations were the result of a coding error by Mathematica.

Under IMPACT, all teachers are evaluated based on classroom observations and other metrics. The value-added formula accounts for 35 percent of the evaluation for teachers in affected grades and subjects.

Teachers are given one of five ratings — ineffective, minimally effective, developing, effective or highly effective. Those rated ineffective are subject to dismissal. The same is true for those rated “minimally effective” two years in a row or “developing” three years in a row.

Of the 22 teachers whose ratings are being raised, Kamras said, three are moving to the highly effective rating; 12 to effective, six to developing and one to minimally effective. The latter is the teacher who was fired mistakenly.

In October, researchers from the University of Virginia and Stanford University who have examined IMPACT reported that its rewards and punishments were shaping the school system workforce, affecting retention and performance. The study found that two groups of teachers were inspired to improve significantly more than others: those who faced the possibility of being fired and those who were on the cusp of winning a substantial merit raise.